Transfer of nonsorbing solutes to a streambed with bed forms:

Transcription

1 WATER RESOURCES RESEARCH, VOL. 33, NO. 1, PAGES , JANUARY 1997 Transfer of nonsorbing solutes to a streambed with bed forms: Theory Alexander H. Elliott x and Norman H. Brooks W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena Abstract. A theoretical analysis of the exchange of nonsorbing solutes between a permeable streambed and the overlying water of a stream or river is presented in this paper. In a companion paper [Elliott and Brooks, this issue] the results of experimental studies of such exchange are presented. The analysis focuses on water movements into, within, and out of the bed which result from the presence of bed forms (ripples and dunes) and highlights the mechanisms of "pumping" and "turnover." Pumping is the movement of fluid through the bed induced by steady dynamic pressure variations over bed forms. Turnover occurs as moving bed forms trap and release interstitial fluid. The detailed analysis was used to generate the distribution of the residence time of solute within the bed, which can be used to calculate the net mass exchange due to pumping and turnover. 1. Introduction patterns inside a porous bed covered with stationary triangular two-dimensional bed forms. In their model, steady pressure The bed of a river can act as a sink or source of contamivariations at the surface of the bed give rise to flow within the nants: pollutants may enter the bed, be stored there for some bed, which was modeled as a uniform porous medium with time, and then be released slowly back to the water column. It two-dimensional flow. They argued that such advective moveis of interest to characterize these exchange processeso that ments cause more bed-stream exchange than diffusion. They their contribution to the contamination and recovery of a river did not, however, calculate the exchange which results from the system can be assessed. flows. Shum [1993] modeled the exchange of reactive solutes This paper presents the theoretical analysis for the case of into a rippled bed under osci!latory water waves, where nonconservative solutes which do not react wkh or sorb to the closure of oscillatory flow paths within the bed gives rise to a solids within the bed or in the water column. However, several net advective transport. of the principles developed in this paper can be applied to the In this paper we attempt to predict the exchange resulting bed exchange of reactive or sorptive substances. An extension from the presence of bed forms in a steady flow by analyzing in of this analysis to adsorbing metal ions is given by Eylers [1994] detail the physics of the mechanisms of exchange. The exandeylers et al. [1995], and the simplified forms of the models change induced by bed forms can be calculated on the basis of presented in this paper have been applied to assess the effect of knowledge of properties of the flow and bed such as stream bed forms on bcnthic oxygen demand [Rutherford et al., 1993, velocity, bed form size, and bed permeability. 1995]. Both stationary and moving bed forms are considered in this Several authors have modeled the exchange processes be- paper. "Bed forms" as used herein refers to ripples or dunes as tween a riverbed or streambed and the water column. The described by Vanoni [1975]. The exchange for other types of exchange is usually treated as a first-order linear exchange bed forms described by Vanoni (flat beds, antidunes, and between the bed and water column compartments [Bencala, chutes and pools) is not addressed. 1984; O'Connor, 1988; Yousef and Gloyna, 1970], as a vertical Bed forms influence the exchange of solutes in two ways. one-dimensional diffusion-advection process ICefling et al., First, when bed forms such as ripples or dunes move, intersti- 1990; Richardson and Parr, 1988; Wallach and Van Genuchten, tial water is released from the upstream face of the bed form 1990; Gschwend et al., 1986], or as a linear time series process where local scour occurs; at the same time stream water is [e.g., Castro and Hornberger, 1991]. In most cases the exchange trapped in interstitial spaces as sediment is deposited on the parameters are determined empirically by fitting to either field lee face. Such exchange occurs even when there is no net or laboratory exchange data. O'Connor[1988] suggests that the erosion or deposition of the sediment. This mechanism will be linear exchange coefficient should be the same as the exchange referred to as "turnover." Second, the acceleration of streamcoefficient across the boundary layer in the flow above the bed. flow over bed forms and separation of the flow at the crest Richardson and Parr [1988] relate the diffusion coefficient to gives rise to spatial variations of the steady component of the parameters such as flow velocity but only for fiat-bed conditions. pressure at the bed surface. The pressure variations induce Savant et al. [1987] have modeled and measured the flow flow into and out of the bed. This process has been observed by Savant et al. [1987] and will be referred to as "pumping." tnow at Department of Natural Resources Engineering, Lincoln The models for exchange of solutes when bed forms are University, Canterbury, New Zealand. stationary are based on the process of pumping. In these mod- Copyright 1997 by the American Geophysical Union. els the interstitial flow inside the bed and through the bed Paper number 96WR surface is calculated on the basis of two-dimensional Darcy /97/96WR flow with prescribed pressure at the bed surface. Then the 123

2 124 ELLIOTT AND BROOKS: TRANSFER OF NONSORBING SOLUTES, THEORY solute flux into the bed surface and the time which the solute spends within the bed (residence time) is calculated. This then gives a transfer function which can be used to calculate the net mass transfer into the bed. change, so thathe effect of bed forms can be assessed readily, and the exchange resulting from bed forms can be compared with the exchange for other processes if the exchange due to the other processes is known. A simple idealized model with a sinusoidal variation of pressure applied above a fiat bed was developed for the pumping process for stationary bed forms. This simple model displays 2. Outline of the Residence Time Function the most important features of the pumping process and can be Approach for Calculating Mass Transfer used to provide analytical or semianalytical approximations of the interstitial velocities within the bed, the flux into the bed, 2.1. Overview and the net mass exchange. This idealized model is extended to regular triangular bed forms, using a numerical simulation and the measurements of Fehlman [1985] of the nonsinusoidal head distribution over triangular bed forms. The effect of stream slope (which gives rise to horizontal underflow) is also considered. Models for the exchange with moving bed forms are also The residence time function approach can be thought of as follows: 1. A pulse of solute enters the bed. The amount of solute entering in this pulse is related to the concentration in the overlying water and the rate at which fluid enters the bed (volumetric flux). 2. The pulse of solute moves through the bed, and some presented with the presumption of uniform two-dimensional may move out. The fraction of the pulse which remains in the shapes advancing at constant speed. In general, both turnover bed a time r after the pulse enters is termed the residence time and pumping contribute to bed-stream exchange of solutes function, R (r). when the bed forms move. However, limiting cases of negligible turnover or negligible pumping can be identified. The limiting case in which pumping is negligible in relation to turnover is referred to as "pure turnover." This limiting case can be treated analytically for triangular bed forms. Further, an approximate solution for pure turnover with random bed forms is presented. In this paper the combined processes are also analyzed for a moving train of regular triangular bed forms. This study focuses on the exchange processes related to bed forms. In the field situation other exchange processes may also operate: 1. Harvey and Bencala [1993] have observed the flow of a 3. A continuous stream of solute entering the bed can be considered as a series of pulses: the total mass remaining in the bed is then calculated by integrating the mass remaining from the individual pulses. This approach is similar to the transfer function model proposed by Jury et al. [1986] for modeling the transport of solutes through soil. To calculate the flux into the bed and the residence time function, detailed models of the flow into, within, and out of the bed were used. The particulars of the flow model depend on the processes being considered and the level of complexity or detail being considered. tracer into and out of the banks of a mountain stream, and they observed piezometric gradients related to this flow in the 2.2. Flux Into the Surface banks. They have presented a simple model for the flow into The inward flux of solute at a point on the bed surface is and out of the banks resulting from variations in water surface denoted by q C, where C is the concentration of solute in the slopes which are in turn related to topographicontrol such as water column directly above the bed and q is the volume flux stepped-bed units in mountain streams. into the bed (equivalento a velocity). In most situations the 2. Groundwater recharge from a stream will result in net solute will be well mixed in the water column, so C could be the exchange, as will groundwater discharge from a previously concentration at any depth in the water column. To simplify contaminated bed. the analysis, we assume a two-dimensional bed surface profile. 3. Shimizu et al. [1990] and Nagaoka and Ohgaki [1990] That is, there is no lateral variation in the bed surface elevation have observed rapid exchange to a depth of a few grain diamnor in other properties of the flow. Further, the diffus, ive flux eters in a gravel bed, which probably results from the penetra- into the bed surface is neglected, as justified through detailed tion of turbulent fluctuations into the bed. While Zhou and discussion by Elliott [1990]. Mendoza [1993] have recently made progress in modeling this The flux into the surface then takes place by two principal flow situation, it is still not possible to predict the associated mechanisms: flow of pore water into the bed surface and trapexchange without calibration of exchange parameters. ping of interstitial water in the advancing face of a bed form 4. Rutherford et al. [1993] suggesthat there can be rapid (pumping and turnover). In the coordinate system of Figure 1 exchange of pore fluid into the fluidized bed material at the crest of dunes. 5. From field observations, Whimzan and Clark [1982] suggest that thermal convective motions (both diurnal and seasonal) are a significant exchange mechanism. 6. Evapotranspiration, bank storage resulting from changes in stream stage, diversion of flow through preferential flow paths in the bed, and flows within the bed resulting from That is, stream curvature are further processes which could result in exchange [Han,.ev and Bencala, 1993]. Clearly, a host of exchange processes can operate in a stream. The role of the theoretical analysis in this paper is to provide insight into the mechanisms of pumping and turnover associated with bed forms. The theory also quantifies the ex- the volumetric flux into the surface at x is q(x)... u n+ 00t Os u n+ 00t Os > 0 (!a) q(x) =0 u-n+ 0 <0 (lb) O q ax a T Ox q(x) = U ss- V s +0 at as arl ax orl Ox -->-0 + o 0-7 as (2a)

3 ELLIOTT AND BROOKS: TRANSFER OF NONSORBING SOLUTES, THEORY 125 Orl Ox Orl Ox q(x) = 0 u - v -s < 0 (2b) where 0 is the bulk porosity of the bed, u = (u, v) is the interstitial seepage velocity vector (0 times the pore velocity), and n is the unit vector normal to the bed surface into the bed. In the case of a bed surface which propagates downstream with a celerity Ut, without changing shape, 0 at = -U Ox (3) The total inward rate of mass flow into the bed over a plan area a of the stream is a C0 where F::/, the average flux into the surface per unit distance down the stream, is 0= qds (4) and A is the bed surface corresponding to the plan area a Residence Time The residence time within the bed is estimated as follows. First, the residence time for solute which enters the bed at a particular point on the bed surface is determined. Then this is averaged over a large area of bed surface (or a representative portion such as a bed form wavelength). The fraction of solute molecules which entered the bed at Xo and remains in the bed after an elapsed time r is denoted by R(xo, r). It is assumed that R is independent of the time at which the particles entered the surface. In the case of steady flow through the bed and no diffusion within the bed, R = 1 before a fluid particle leaves the bed and R = 0 after a fluid particle leaves the bed. The average residence time function, (r), denotes the fraction of solute which entered the bed in a short time near t = 0 and remains in the bed at time r. Since the flux into the surface varies with position, R (Xo,,) must be weighted by q to determine the spatially averaged residence time function, so with r. I qr(xo, r) ds fa qr(xo, r) ds q ds = = ao (5) bed surface X,U \11 S Figure 1. Definition sketch of geometry; '1 is the elevation of the bed surface above the mean bed surface, s is the distance along the bed surface, x and y are the spatial coordinates (horizontal and vertical), u and v are the seepage velocities (horizontal and vertical), and n is the unit vector normal and into the bed surface. or m(t)co = [t -0 R(?)C(t -,) dr re(t) = c-1 R(r)C*(t- r) dr '-0 where m is the accumulated mass per unit plan area of stream divided by a reference concentration Co, and C* = C/Co. In the special case of a step change in concentration from 0 to Co at t= 0, (8)becomes (7) (8) re(t) = 0 (,) dr (9) t The quantity m has the dimensions of length. If the solute were mixed uniformly into the bed to a depth m/o and the concentration in the fluid to that depth were Co, then the accumulated mass per unit area ofbed would be mco. That is, m/0 can be considered to be an effective mean depth of mixing into the bed. To calculate the mass exchange into the bed, the time history of concentration in the water column must be known. This could be measured in the field or calculated by coupling (8) for the bed exchange with a conservation equation describing the Shortly after a pulse of solute enters the bed (small r), most longitudinal transport of solutes in the streamflow. of the pulse of solute remains in the bed, so tends to 1 for Detailed models were used to determine q and R in various small r. As time progresses, some of the pulse of solute exits flow situations. The analysis for stationary bed forms will be from the bed. Therefore the remaining fraction ( ) decreases presented first, followed by the analysis for moving bed forms Net Mass Transfer The mass transfer which results from a given history of concentration in the overlying water is calculated as follows. The mass of solute which entered the bed over a small time d r at a past time t - - is OC(t - r)dr per unit plan area of the streambed. A fraction R (r) of this remains in the bed at a time t. Thus the incremental contribution to the mass within the bed at time t from flux into the bed at time t - r is OR(r)C(t - z) dr (6) Therefore the accumulated mass in the bed (considering all elapsed times r) is 3. Analysis for Stationary Bed Forms 3.1. Sinusoidal Pressure Applied Above a Flat Bed First, an idealized case of a sinusoidal variation of pressure applied above a flat bed is considered. The bed material is considered to be homogeneous and isotropic. In this idealized case, analytical forms for the volume flux into the bed (q) and the residence time function (R) can be found. In other cases, numerical simulations are used to calculate q and R. The idealizations of a sinusoidal head variation and a flat bed have the following basis. First, the pressure distribution over the surface of the bed (Figure 6) can be approximated by a sinusoidal curve. Second, the vertical component of pumping

4 126 ELLIOTT AND BROOKS: TRANSFER OF NONSORBING SOLUTES, THEORY 1.0- u = --Urn COS kxe ' (13a) h m v = -urn sin kxe ' where u,,, the maximum velocity, is (13b) Um= Kkhm (14) [ i i I I i Normalization of quantities. The velocity field for the sinusoidal head problem suggests the following normaliza. tion for space and time: x* = kx y* = ky (15a) t*/o = k2kh,,t/o (15b) where 0 is the porosity of the bed material. The timescale can be interpreted as follows: in time t*/o - 1 a fluid particle traveling at the maximum pore velocity (u,,/o) will travel a distance X/2rr. The spatial variables are normalized with respect to X/2,r (that is, the inverse of the wavenumber). The velocity distribution is then normalized as follows: dx * u u u*... d(t*/o) Kkhm u, (15c) Figure 2. Normalized head distribution (h/it,,), streamlines (solid lines) and front positions (dashed lined) for the sinusoidal-head model. The fronts are shown at time t*/0 = 2.5, 5, 10, 20 (deeper fronts correspond to larger time). dx * v v v* = d (t* / 0 ) = Kkh,,--- = u,,-'- (15d) The spatially averaged volumetric flux into the surface, can also be normalized by u m = Kkhm, so F:/* - (16) The formula for the mean residence time function (equation flow into and out of the bed, which is the most important, is (5)) can then be expressed in terms of normalized variables by retained if the bed becomes flat (while retaining the same pressure distribution over the bed). Finally, the flow patterns within the bed (especially deeper in the bed) are little affected (17) a =,;,f Rq*ds*,4 * by the topography (again assuming the same pressure distribution at the surface). As shown later, these approximations where a* = ka, s* = ks, and,4' = k,4. result in only minor errors (see Figure 7) in relation to the case The accumulated mass transfer is m Co, so m has the dimenwith measured (nonsinusoidal) head applied over triangular sions of length and is normalized as follows: bed forms. Further, Rutherford [1994] and Rutherford et al. [1995] have illustrated the similarity of the streamlines for the idealized case as compared with the triangular-bed form case Velocity field for the sinusoidal-head model. Conm* = 2,rkm o sider a sinusoidal variation in pressure applied over a flat level bed (see Figure 2). The dynamic head variation at the bed where 0 is the porosity. This is essentially a normalization by a fraction 0/(2'n') 2 of the wavelength. Since k = 2 -/X, surface is given by hk,_-0 = h,,, sin (kx) (10) The solution for head and porewater seepage velocity is then t = h,,, sin kxe " (12) m m* Xm* - = X (2 r)2 40 (19) where hm is the amplitude of the head variation (total variation = 2h,,, as shown in Figure 2; see (28) to evaluate hm) and With this normalization, (8) becomes k is the wavenumber of the variation (k = 2 r/x, where X is the wavelength of the bed forms). m* =2 rf:/* The pore water seepage (Darcy) velocity is given by u = f,:' C* (t* -- ) d (2O) (_ _) (u, v) = -KVh, where K is the bed hydrauliconductivity. For a homogeneous isotropic bed, h is then given by the solution of Laplace's equation where C* = C/Co and Co is a reference concentration which is constant over time. V2h = 0 (11) Residence time and mass transfer for the sinusoidal-head model. Appendix A contains details on the deriva- tion of the streamlines, front positions, flux into the surface, and residence time function for the sinusoidal-head model. Figure 2 shows the streamlines and front positions.

5 , ELLIOTT AND BROOKS: TKANSFER OF NONSORBING SOLUTES, THEORY I t */e Figure 3. Mean residence time function,, for the sinusoi- dal-head model versus normalized time. The spatial!y averaged flux into the surface is given by (A8), and the normalized flux is given by (A9): Urn Kkhrn -- - (21a) 1 F::/* = -- (2lb) - = (21c) where is obtained from (2!c). For this case the mass transfer m * is shown as the upper curve (curve e) in Figure 4 and also as the uppermost curve in Figure 5. The initial increase mass is rapid, followed by a slower increase. At t* / 0 = 1000, m * is equal to approximately 40. From (19) the solute will have then penetrated to an average depth (m/o) approximately equal to X. At that time and thereafter the rate of increase of depth of penetration is very small in relation to the initial rate of in- crease. For small time nearly all the solute which enters the bed remains in the bed. That is, R tends to 1 as t tends to 0. Therefore the mass exchange is approximately 2t* m* = (23) 0 Figure 3 shows as a function of t*/o. The accumulated mass The asymptotic solution for large time has also been detertransfer can then be calculated from (20). The integration is mined: performed numerically. m* = 2st In (0.42t*/0 + 1) (25) In the case of a step change in concentration t = 0 (see (9)) the normalized mass transfer for the sinusoidal-head This gives a good approximation to the full solution for t* / 0 > model reduces to 3. Note that the net mass exchange does not approach a finite limit as time becomes large for this model. m*=2 ' 0 d (22) This approximation is good only for t*/o < 1 [Elliott, 1990]. Moreover, for t*/o < 1 diffusion into the surface [Elliott, 1990] can affect the net mass exchange, and other rapid exchange processes [e.g., Nagaokand Ohgaki,!990; Rutherford et al., 1993] may be dominant, so care should be taken in applying (23). For 2 < t*/o < 25 the solution to the sinusoidal-head model is approximately and the residence time function, R (t* / 0 ), can be determined m* - 3.5(t*/0) u2 (24) from the following implicit relation (equation(a13)): This approximation has the same form as the solution for t* 2 cos - R vertical diffusion into a semi-infinite bed. This may explain why some investigators have applied a diffusion-type model to their data with some success. The solution for a finite-time concentration pulse, which represents a contamination-recovery cycle, is obtained by sim- 4O i I i 35 3O 25 T*/0 m* 2O 15 b 50 c 100 _ d 4l)0 T 10 o t* Figure 4. Normalized net mass exchange (m*) versus normalized time (t*/o) for a pulse change in concentration in the overlying water. The results are shown for various values of T*/O, the normalized pulse duration. The inset shows the concentration history. The top curve (curve) is for a step change in concen- tration (i.e., r*/o --> o ). e

6 128 ELLIOTT AND BROOKS: TRANSFER OF NONSORBII IG SOLUTES, THEORY m ß '085 I0., I i I I I, I t* 0 Figure 5. Effect of underflow on mass transfer (m*) for a step change in concentration for various values of normalized underflow velocity (u ng)- Increased underflow leads to decreased net mass exchange. ple superposition of the solutions for step changes in concen- height to water depth, as could be expected for dunes), then tration. The concentration of solute in the water column jumps there will be a corresponding increase in S. from 0 to C O at t -- 0, and is then held at C O until time T. The The modeled effect of underflow on the net mass transfer is concentration in the stream then falls to 0 again. The solution demonstrated in Figure 5, which is for a step change of solute is shown in Figure 4 for various values of T*/O. The mass in concentration in the water column at time t* - 0. In this the bed drops rapidly once the concentration in the overlying model the interstitial velocity field was calculated by adding the water drops, but some solute remains in the bed and seeps out underflow to the velocity from the sinusoidal-head pumping of the bed for a long time. model. The residence time function was determined by a 3.2. Effect of a Stream Gradient straightforward numerical particle-tracking procedure (see section 4). The stream gradient induces a nearly horizontal interstitial The effect of increasing the underflow velocity (u l*ong) is to underflow in the streambed. This underflow has an effect on reduce the net mass exchange (see Figure 5). The underflow the residence time distribution, R. Fluid particles enter the bed has little effect on the initial exchange. Eventually, however, in regions where the flow is into the bed. They are then swept underflow causes the net flux to reach zero (the cumulative along with the underflow to regions where the flow is out of the mass transfereaches a limiting value), whereas without unbed. The residence times are therefore shorter than those calculated without underflow, and underflow reduces the net derflow the accumulated mass transfer continues to increase without limit. The limiting value for m * and the approximate mass transfer into the bed. time to reach the limiting rn * both decrease as u * long increases For a stream gradient S the corresponding seepage velocity (see Figure 5). of the underflow is For example, for u * long (which is a typical value), m* u o, g = KS (26) approaches a value of 20. Therefore (from(19)) the effective depth of solute penetration is typically about 0.5 of a bed form This velocity is normalized by u m, the characteristic pore water wavelength. As an illustration of this point, in a stream with velocity due to pumping (Kkhm): bed forms 1 m long and u* tong = 0.085, the model predicts that the solute will penetrate to and average depth of 0.5 m. bl * Ii long S long = u,, - kh,,z (27) Values of u * long determined the laboratory experiments [Eltiott and Brooks, this issue] were 0.04 to Such values can also be expected to apply in field situations [Elliott, 1990, p. 67] in cases where the bed is covered with dunes and form drag dominates the total drag, because factors which increase h,, z and k will also increase S. For example, both h,,z and S increase with U 2. Similarly, if k is increased (while keeping the same bed form aspect ratio and the same ratio of bed form 3.3. Exchange for Regular Triangular Bed Forms (Stationary) The pressure distribution over solid regular two-dimensional triangular bed forms has been measured by several investigators, including l, qttal et al. [1977] and Fehlman [1985] (reported also by Shen et al. [1990]). We used Fehlman's pressure dat as a surface boundary condition for a numerical solution of Laplace's equation for the interstitial velocity within the bed. This flow information was then used to determine q and R. Fehbnan [1985] gave data for the dynamic head distribution

7 ELLIOTT AND BROOKS: TRANSFER OF NONSORBING SOLUTES, THEORY 129 h i I i i X X* 2 2x Figure 6. Normalized piezometric head distribution over a triangular bed form, corrected for the mean hydraulic gradient (derived from Fehlman [1985]). The trough of the bed form is at x = 0. The maximum head variation, hm, can be calculated from (28). for solid triangular bed forms (height/wavelength = H/X = 1/6.7). Further, an empirical expre.ssion for the variation in form-drag coefficient with the ratio of bed form height to mean water depth (H/d) was given. From this an empirically based expression for the half amplitude of the dynamic head variation, hm, can be derived: S 2 hm = O-] H/d J 3/8 H H/d 3/2 m/a / d _< 0.34 >_ 0.34 (28) where U is the mean stream velocity. The head and velocity distribution inside the bed was calculated using a finite element numerical solution of Laplace's equation [Elliott, 1990] with the head distribution shown in Figure 6, which is derived from Fehlman's data. Triangular bed forms with H/X = 1/7 were used, and an impermeable bound- ary was placed at a depth of y = -X. A periodic boundary condition (equal head) at x = 0 and x = X was used. The calculation of the mass exchange proceeded by placing a number of fluid particles (typically 1600) with even spacing (in terms of x) at the surface. Each fluid particle was representative of solute entering the bed through ds*, an element of the surface. Each particle was given a weight q'ds*. The quantity F: t* was then calculated by summing the weights for all the particles and dividing by a*, the sum of the horizontal components of the bed surface elements. The particle paths through the bed were calculated using the velocities determined in the numerical solution of the flow and Runge-Kutta integration of the velocity. At each time step (typically Xt*/O = 0.05) the new positions of all the particles were found and (t*/o), the residence time function at time t*/o, was calculated by summing the weights for all the particles still in the bed and dividing by/:/*a*. The mass, rn*, was then determined for a step change in concentration t* = 0 by straightforward numerical integration of (20) (with C* = 0 before t = 0 and C* stepping from 0 to 1 at t - 0). The results for u* long = 0 are shown in Figure 7. The exchange in the triangular bed form model is slightly greater than the exchange in the sinusoidal-head model with the same he. The small difference between the net exchange for models can be attributed partly to the triangular rather than fiat bed surface (increased surface area for exchange) and partly to the nonsinusoidal shape of the pressure distribution. The flux into the surface was increased by about 40% (F::/* 1.4/z-) for the triangular model, however. This increase is associated with the shorter-wavelength components of the pressure distribution, which induce higher velocities into the bed (see (14)). These shorter-wavelength components have little influence on the net mass exchange for large time because they only affect the penetration near the surface. 4. Analysis for Moving Bed Forms 4.1. Overview In the analysis for moving bed forms we assume that the bed forms propagate at a constant celerity and that they maintain 351 i i 30 t O I I I ( t* Figure 7. Comparison between the triangular bed form model and the fiat-bed sinusoidal-head model for no underflow. The normalized net mass exchange (m*) for a step concentration change is plotted versus normalized time (t* / 0 ). ' O

8 130 ELLIOTT AND BROOKS: TRANSFER OF NONSORBING SOLUTES, THEORY direction of flow and bedform movement original crest position final crest position..,.:--,,.?: :- ',.::;.- ' ---.< ::g.,... Figure 8. pore water original pore pore water removed water retained deposited Schematic of the idealized movement of a bed form and the resulting turnover exchange process. their shape as they move. We adopt a frame of reference which 1-N moves at Ut,, the speed of bed form propagation. In this frame the bed forms appear stationary while fluid, including pore (32) water, has an extra component of velocity equal to -U,. This simplifies the calculations considerably. In general, both turnover and pumping contribute to the For regular triangular bed forms the depth of penetration is bed-stream exchange when bed forms move. However, two limiting cases can be identified. First, if the bed forms move limited to H/2. That is, the solute does not penetrate deeper rapidly in relation to the pore water, then pumping can be than the bed form troughs. In the natural situation some largneglected and we have "pure turnover." This situation can be er-than-average bed forms exist. As the larger bed forms propagate they trap pore water, so that solute eventually penetrates approached analytically, and it was analyzed for both triangubeyond the depth of the average bed form trough. lar and random bed forms. The second limiting situation oc Pure turnover for random bed forms. A model of curs when the pore water velocity is large in relation to the propagation speed of the bed forms. In this case turnover can turnover for random bed forms was developed, on the basis of be neglected, but the movement of the bed forms should still a random bed surface generated by a Gaussian process [No : din, 1971]. The full analysis using a residence time approach is be taken into account because this causes an unsteady velocity field within the bed. given by Elliott [1990], but here we only present an approximate result which applies once several bed forms have passed Pure Turnover Crickmore and Lean [1962] have shown that for bed forms modeled with a Gaussian process the average depth of scour, Pure turnover for regular triangular bed forms.., is Consider Figure 8, which shows the mechanism of turnover for a triangular bed form. The bed form is originally uncontaminated, and there is a step change in concentration in the water 22/20' Y = (In N) + 53,(In N) - (33) column at t = 0. After time t, only the area bounded by the where 3' is Euler's constant ( ), o- is the r.m.s. bed center triangle remains uncontaminated. The effective depth of surface elevation, and N is the number of bed forms which penetration, m/0, follows: have passed (see(30)). H,X The depth of scour at any point is the maximum depth that m_ y(1-(1-n) 2) t<u-- has been uncovered by bed forms. This is equal to the depth of 0 H h (29) penetration of solute. Hence where N, the fraction of a bed form which has passed, is Ut, U t N-- (30) From the geometry of the advancing bed form face so from (9) H (3 ) Ocr m --.' cr _.. 2 /2[(ln N) + ;3,(In,. 1 N) -1/2] (34) A useful result for determining o- is that o- is approximate15 H/2 for natural bed forms [Nordin, 1971], where H is the mean bed form height. The mass exchange for regular and random bed forms is shown in Figure 9 (where we used o- = H/12 /2 for triangular bed forms). This illustrates that as time becomes large, the turnover exchange for random bed forms becomes larger than the exchange for regular bed forms. Elliott [1990] also showed that for random bed forms,

9 ELLIOTT AND BROOKS: TRANSFER OF NONSORBING SOLUTES, THEORY 131 _ Triangular rn/( e) 2 bedlorm/ _ m* i i i iiiii I i! i iiiiii i i i ii,,l i i i iiiii Figure 9. Predicted average depth of solute penetration after a step change in concentration for "pure turnover" with random bed forms or triangular bed forms of uniform height. N is the number of bed forms which have passed, while m/(cro) is the depth of solute penetration (m/o) divided by the r.m.s. of the deviations in bed surface elevation (c 0. N 0! i i i Figure 10. Net normalized mass exchange (rn*) for combined pumping and turnover for triangular bed forms as a function of normalized time (t*), for various values of the normalized bed form velocity (U, - u *.o_ng)-the limiting depth of penetration for fast bed forms (U, - c ) is half the bed form height. ment are accounted for by adding an upstream component of horizontal velocity equal to Ub - u ong/o in the moving refer- = ¾ (35) ence frame (which moves at the bed form speed, Ub). The particle-tracking calculations were performed in the which is about 25% more than that for triangular bed forms usual fashion (see section 4) using the approximate analytical (equation (3!)). solution for the interstitial velocity field. An aspect ratio of i: Combined Turnover and Pumping for Regular and five velocity components (n = 1 to 5) were used. The results of the calculations for combined turnover and Tdangu!ar Bed Forms pumping with regular triangular bed forms and a step change The limiting case of pure turnover can be expected to apply in concentration in the water column at t* = 0 are shown in when the bed forms move rapidly in relation to the pore water. Figure 10 and Table 1. The table also shows the pumping In general, however, both pumping and turnover may contribexchange for moving dunes without turnover for the same ute to the exchange. The combined processes were simulated U, - u* long' The turnover effect is eliminated by using a flat to calculate the combined exchange as well as to identify the surface but retaining the same moving pressure distribution. limiting cases. The relative magnitude of bed form movement and pres- For (U, - U 'ong ) < 0.5 (approximately) there is little difference between the mass exchange calculated with and sure-induced p.ore water movement can be expressed as a without turnover for triangular bed forms. In this range turndimensionless parameter. The characteristic pumping pore veover can be neglected, but not the effect of the moving pressure locity is U,n/O, and the characteristic apparent horizontal vepattern due to dune propagation. locity is Ub - Utong/O (in the negative x direction). Therefore In general, the steepness of the bed forms will also affecthe the parameter expressing the relative effects of turnover and relative role of pumping and turnover. For pumping the mass pumping is exchange is strongly dependent on the bed form length, X, 0U -u,o. U;-u* /2rn = long (36) Table 1. Effect of Turnover on the Normalized Net Mass It can be anticipated that turnover will dominate the exchange Transfer at t*/o = 100 for Triangular Bed Forms for U, - u* long >> 1, while turnover can be neglected for Normalized Bed Form Normalized Net Mass Transfer m* Velocity, We used an approximate analytical solution for the flow U;- u* long within the bed for this problem. The head distribution of Fehl- With Turnover man [1985] was resolved into Fourier components with asso ciated wavenumbers kn. Then the pressure distribution due to each component was assumed to die off exponentially from the bed surface, r/, giving tz = hm {an sin (k,x) exp [k (y- r )] n=!,2,.. ß Large b,, cos (k,;c) exp [k,,(y- '!)]} (37) Without Turnover Turnover has little effect relative to pumping for U - u 7o,g < 0.5, where a n and b, are the Fourier coefficients. The velocity while turnover is dominant when the bed forms move rapidly in relafollows from Darcy's law. The underflow and bed form move- tion to the pore fluid (U} - u* tong > 10).

10 132 ELLIOTT AND BROOKS: TRANSFER OF NONSORBING SOLUTES, THEORY while for turnover the exchange increases with bed form height,/-/. Therefore for the combined processes the parameter H/X must be relevant. We have not investigated the effect of this parameter. We postulate that the result for H/X = 1/7 can be extended X/d a (bed form wavelength/bed grain size) = 180. The effect of dispersion decreases as this ratio increases. In our flume experiments [Elliott and Brooks, this issue] the ratio was always greater than this, so longitudinal dispersion could be neglected. By contrast, the transverse component of dispersion can for arbitrary aspect ratios (H/X) by comparing the character- result in a significant increase in mass exchange. This arises istic flow into a bed form due to pumping to the characteristic because transverse dispersion can transfer solute from shallow flow due to turnover and underflow. The characteristic flux streamlines (short residence times) to streamlines which run into the bed due to pumping is u,, so the flow into a bed form is characteristically u, X. The flux due to turnover and underflow is (OUt,- U ong), but this inflow occurs only over the deep into the bed (long residence times). For example, with X = 180dg and a lateral pore-scale dispersion coefficient equal to 0.2 times the longitudinal component, the transfer was in. height of the bed form. The flow into a bed form is therefore creased by 30% at t*/o (large time). Similarly, H(OUt, - Utong)- The ratio of these flows is (H/X)(U - ut*ong ). exchange was increased by 30% for ODm/Kh,, z = at t*/0 = In most cases the simulated effect of dispersion Using the simulation result for H/X = 1/7, the generalized condition is that bed form turnover can be neglected for ( /X)(V; -,'o ) < was less than this. The predicted mass exchange decreases with increasing 5.2. Large-Scale Variations in Pore Water Velocity U, - u* long up to a value of U} - u* long 2. Fluid which is Large-scale variations in pore water velocity may result eiswept into the bed is swept out of the bed when the bed forms have moved half a wavelength and the velocity field has reversed. Hence the faster the bed forms move, the shorter is the residence time and the smaller the mass exchange. Furthermore, fluid which enters the upstream face of the bed form may have only a short residence time because the bed is being eroded on that face. The faster the bed forms move, the more likely it is that turnover will interfere with pumping. This exther from large-scale irregularities in the pressure at the bed surface or from large-scale longitudinal heterogeneity in the bed permeability. The pressure anomalies may result from irregularities in the bed surface, obstructions in the bed, bends in the river, or changes in the channel width. With large-scale permeability anomalies, flow must enter the bed to supply the increased underflow in areas with higher permeability and fl0w leaves the bed in zones of lower permeability; this phenomeplains why the total mass exchange is less than the exchange non increases the overall exchange of water and tracer. calculated without turnover in some cases. We can conceptualize these flows as being driven by large- The predicted exchange for combined processes with scale head variations at the surface of the bed. Equation (14) (U, - u Tong) = 2 is less than the exchange for either pure can be used as a basis for relating the amplitude and scale of turnover or pumping without turnover. This is counterintuitive, the pressure variations to the variations in pore water velocity. as it might be expected that the exchange for combined pro- Elliott [1990] analyzed the effect of large-scale variations by cesses would be greater than the exchange for the separate particle tracking, using a flow field created by the large-scale processes. In fact, the results predict that the processes disrupt head variations superimposed on the usual sinusoidal head each other. variation. The effect of the large-scale variations is to increase As U, - u* ong increases beyond 2, turnover starts to have the mass exchange for large time. For example, for anomalies a dominant effect. For (U - u ong) > 7 the exchange is close with a scale of 10X and a pressure anomaly of +0.5hm, the to the exchange for pure turnover, and pumping can be ne- mass exchange is increased by 50% at t*/0 = This glected. This corresponds to (H/X)(U} - u 'ong) > 1. Since increase occurs because the variations introduce deeper u* long is usually less than 0.25, this condition is essentially streamlines (associated with longer residence times). H/X U, > 1. This result gives only a rough indication of when Further, there could be preferential flow paths within the turnover dominates in natural streams because, as demonstrated earlier, a model with regular triangular bed forms provides only a rough indication of turnover. bed, for example, high-permeability flow paths running along old channels which have now been covered by sediment. The analysis of such flow is beyond the scope of this paper Random Bed Forms With Pumping 5. Extensions to the Analysis The inclusion of a random component to the bed forms (as Several extensions were made to the analysis, which are opposed to regular triangular bed forms) can be expected to summarized briefly here. Further details are given by Elliott increase mass transfer in two ways. First, it will increase turn- [ over, as already demonstrated in section (pure turnover for random bed forms). This occurs because the occasional 5.1. Molecular Diffusion and Pore-Scale Dispersion large bed form scours the bed to a large depth and then traps The idealized model for stationary bed forms (sinusoidal stream water until another large bed form comes. Second, head variation) was extended to assess the effect of molecular random bed forms introduce large-scale variations in the pore diffusion and pore-scale dispersion within the bed [Elliott, velocity, which increases mass exchange (see section 5.2). 1990]. Both a longitudinal and lateral component of the pore- Simulations by Elliott [1990] demonstrated that the randomscale diffusion were considered, using values based on a study ness increases the exchange. For situations where the bed by Bear [1972] and the predicted pore water velocity. Disper- forms are stationary or U is small, the randomness may insion was incorporated into the numerical residence time cal- crease the net exchange by typically 35% for t*/0 = 500. This culations through a random walk particle-tracking procedure effect varies depending on U; - u* long and the degree of [Bear and I,'emtijt, 1987]. variability of the bed forms. The model also indicates that the The simulations demonstrated that longitudinal dispersion randomness increases the flux into the surface by approxihas an insignificant effect on the predicted mass exchange for mately 30% as compared with the sinusoidal-head flat-bed

11 ELLIOTT AND BROOKS: TRANSFER OF NONSORBING SOLUTES, THEORY 133 model. Pumping for random bed forms was dominant for fluid velocity, the mass exchange occurs only as the moving bed (H/X)(U;- UTong) < 0.03, while (H/X)U; needs to be form scour and deposit sediment and pore water (turnover). increased to more than 3 for turnover to be dominant. In the absence of net erosion or accretion the depth of solute penetration (or solute removal) is limited to the depth of bed 6. summary of Models and Discussion of scour. For regular triangular bed forms the analysis of turnover is simple but has shortcomings because the depth of scour is Applicability limited to the constant depth of bed forms (whereas for natural In this paper various physically based models are presented bed forms the scour depth slowly increases the number of to attempt to predicthe effect of bed forms on the exchange bed forms which have passed increases). For pure turnover the of solutes between a streambed and the overlying water. The principal scale for the depth of solute penetration is the bed models in this study require only a knowledge of the flow in the form height (triangular bed forms) or the standardeviation of stream and bed properties to calculate the exchange resulting the bed surfacelevation (random bed forms). The scale for from the presence of bed forms. time is the time it takes for a bed form to propagate a wave Models for Stationary Bed Forms (Pumping) length Combined pumping and turnover. For the com- The flow over irregularities in the bed surface (such as rip- bined processes of pumping and turnover with regular trianples or dunes) gives rise to spatial variations in the steady gular bed forms it was found that turnover can be neglected for component of pressure at the bed surface. These variations in sufficiently small bed form celerity (U, - u * long < 0.07X/H). turn give rise to flow into and out of the bed, which results in For regular triangular bed forms with an aspect ratio of 1:7, mass exchange. This process is termed "pumping." pumping can be neglected provided U; > 7. This corresponds Sinusoidal pressure distribution applied over a fiat to (H/A)U > 1 for a general aspect ratio. bed. For the idealized case of a sinusoidal pressure variation applied over a fiat bed, an analytical solution for he flow and mass exchange due to pumping was found. This simplified 6.3. Discussion of Applicability of the Models In the field (and even in the laboratory) processes such as model captures the main features of pumping wkh stationary groundwaterecharge/discharge, large-scale variations in bed bed forms. The model has also been adapted for sorbing sol- topography (such as bends and pool/riffle sequences), bank utes [EyIers, 994] and for oxygen-demanding substances storage, thermal convection, preferential flow paths within the [Rutherford et al., 1995]. Initially, all solute which enters the bed remains there, and the mass exchange increases linearly with time (t) for a step increase in concentration in the overlying water at t = 0. Later some solute leaves the bed so the net flux into the surface is reduced. In the period 1 < t*/o < 25, the net mass exchange increases with t 1/2. At later times the mass increases with the logarithm of time and does not reach a constant limit; some of the solute which enters the bed never emerges. bed due to heterogeneity of the bed material, and rapid exchange very near the surface due to turbulent pressure fluctuations could all influence the exchange in addition to the exchange due to bed forms (as discussed in more detail in the introduction). We agree then with Bencala et al. [1993, p. 183] that "modeling the fluid mechanics in the hyporheic zone is a formidable task," because there are many processes with range of temporal and spatial scales with natural variability. Some suggestions Results for the pumping process were expressed in dimen- for an approach to comparing the processes are given in secsionless form. The principal scale for the depth of solute penetration is the bed form wavelength and the scale for time is essentially the time for pore water to travel a bed form wavelength Effect of longitudinal underflow. A uniform undertion 6.4 of Elliott and Brooks [this issue], but we have not attempted to provide a detailed comparison of the timescales and length scales which operate in natural streams. Even if we are unsure about the importance of bed formrelated processes in comparison to other processes in the field, flow induced by the stream gradient does not alter the mass are we at least confident that exchange related to bed forms exchange for small times but eventually causes the net flux into can be predicted adequately? The laboratory experiments [Elthe surface to approach zero. The average depth of solute liott and Brooks, this issue] indicate that pumping flows related penetration (or net mass exchange) then approaches a finite limit, which is typically half of the bed form wavelength Triangular bed forms. A numerical simulation of to bed forms are modeled satisfactorily (at least for stationary bed forms) and that pure turnover is modeled adequately. However, the models significantly underpredicted the net acexchange with regular triangular bed forms (with measured cumulated mass exchange for large time. The flow visualization pressure variations from Fehlman [1985]) gave a net mass and analysis of mass exchange suggest that much of the addiexchange close to the exchange for the idealized case of a tional mass exchange in the experiments results from processes sinusoidal pressure variation applied over a flat bed, although not related to bed forms (such as interstitial flows driven by the volumetric flow rate into the surface is greater than for the steady irregular pressure variations at the bed surface), which sinusoidal-head case. For practical purposes the simpler model were not included in the models of exchange related to bed is adequate. forms. Therefore we have some confidence that the models of Model extensions. The modeling by Elliott [!990] pumping and turnover provide a reasonable first-order estidemonstrated that dispersion, large-scale irregularities in the mate of exchange related to bed forms. pore velocity, or randomness in the bed forms increase the mass transfer Models for Moving Bed Forms Clearly, the predictive approach based on detailed modeling of the underlying processestill has some way to go before the exchange in the field can be predicted confidently. Nevertheless, the analysis presented in this paper has the following Pure turnover. In the limiting case, when the speed useful aspects: of advance of the bed forms is rapid compared with the pore 1. The analysis gives insight into the processes and length

12 134 ELLIOTT AND BROOKS: TRANSFER OF NONSORBING SOLUTES, THEORY scales and timescales of solute exchange resulting from the so presence of bed forms and can be used to estimate the ext* change resulting from pumping and turnover. This can provide x* = x; - ¾ cos x; lower-bound estimates of the total exchange, can be used to (As) compare to estimates of exchange for other processes, and may provide a basis for interpretation of field results. 2. Various simplifications of the models have been develcos x;- - cos x; oped and compared to more detailed models. This provides Y* = -ln cos X (A6) insight into the level of detail necessary to model pumping and turnover. 3. Simplified versions of the models have already been extended with some success to studies of the exchange of adsorbing metal ions in the laboratory [Eyters, 1994; Eylers et al., 1995] and of benthic oxygen demand in the field [Rutherford et al., 1993, 1995]. 4. The modeling approach is based on analysis of the un- The front positions, that is, the locus at time t of fluid particles which were at the bed surface at t = 0, are then given parametrically in terms of the parameter X ) by (A5) and (A6). Front positions are shown in Figure 2. We now calculate the spatial distribution of flow into the bed. Considering X ) from -,r to,r, the volume flux into the surface follows from (13b) and (1): derlying processes, which offers an alternative to approaches based on well-mixed compartment models, diffusion models, or general linear time series (compartment and time delay) models, where exchange parameters determined for one q= u. sin kx 0 < X < sr (A7) stream cannot generally be used in another stream. 5. The well-known concept of transfer function or resi- The spatial averaging of q can be performed over one wavelength, so from (4) with a =,4 = X, dence time function was applied to the calculation of net stream-bed mass exchange for an arbitrary history of concentration in the overlying water, which complements transfer q - = --,r (A8) function or time series interpretations of field data [e.g., Castro and Hornberger, 1991]. 6. This study may serve as a precursor to further fundamentally based modeling studies of bed-stream exchange. The streamlines are then given by dy* v* dx---; = u* = tan X* (A2) cos Y* = -ln co sx (A3) where Xo is the position of the streamline as it passes through the bed surface. Typical streamlines are shown in Figure 2. Consider a fluid particle which enters the bed at Xo and time t*/0 = 0. The subsequent position of the particle can be found from the velocity field and the streamline positions. On a streamline, from (AI) and (A3), dx* d(t*/o = -cos X; (A4) and by (16) the normalized flow into the surface is 1 F/* = -- The residence time function for the sinusoidal head problem Appendix A: Details of Analysis for Pumping may now be determined. From (A5) a fluid particle which With a Sinusoidal Head Applied enters the bed at X* r between 0 and sr/2 and at t = 0 will exit Over a Flat Bed from the bed at -X* r at time The conceptual basis of the sinusoidal-head model and the derivation of the associated pore water velocity distribution is 2X c*r presented in section 3.1 as is normalization of various quanti- t*/0 = cos (X* r) (AI0) ties. This appendix presents further details on the streamlines, front positions, volumetric flux into the surface, and residence At this time (t*/0 ) all particles which entered the bed between time function. The normalized velocity distribution (from 0 and X*cr at t = 0 have already exited the bed as have particles (13)-(15)) is which entered between,r - X* r and sr. Hence the residence time function can be determined: u* = -cos x*e? v* = -sin x*e y' (A1) n = 0 0 _< x; _< x;, - X*c -< x; -< r (^ a) We can now calculate the streamlines and front positions. Let X*, Y* be the normalized location of a streamline. From the R = 1 X*cr-< X; < sr - Xc*r (Allb) velocity distribution R does not apply if -sr --< X ) --< 0 (since q = 0). By (A7), (A9), (17), and (All), the mean (spatially aver- aged) residence time function becomes (A9) R(t*/O) = 2,r(1/ ') sinx dx = cos Xc*r (A12) With (A10) and (A12) an implicit relation for can be found: t* 2 cos - ¾ = (A 3) This relationship, shown in Figure 3, is used in the main text to determine the net mass exchange for the sinusoidal-head model (following(21)).

13 ELLIOTT AND BROOKS: TRANSFER OF NONSORBING SOLUTES, THEORY 135 Notation A, a surface and plan area of the bed (respectively) for integration, m 2. C solute concentration in the overlying water, mg/l. Co reference concentration used for scaling of C and m, usually taken to be the initial concentration in the water column, mg/l. C* normalized solute concentration, equal to C/Co. d mean water depth, m. db bed depth, m. d e geometric mean grain diameter, m. H bed form height (trough to crest), m. h dynamic head, m. hm amplitude of the dynamic head fluctuations the bed surface (total head variation hh,,,), m. K hydrauliconductivity of the bed, m/s. k wavenumber, usually bed form wavenumber (2,r/X), --I m. m mass transfer per unit bed area, divided by Co, m. m* normalized value of m, equal to 2 rrkm/o. N fraction of a bed form or number of bed forms which have passed. q mass flux into the surface, divided by C, m/s. q* normalized flux, equal to q/u,,,. g/ flux (q) averaged over the bed surface, m/s. R residence time function. flux-weighted spatial mean value of R. S hydraulic gradient or stream slope. s distance along the bed surface, m. t time, s. t* normalized time, equal to k2kh,,,t. U mean flow velocity in the channel, m/s. U., bed form propagation speed, m/s. U, normalized bed form propagation speed, equal to OU,/u,,. u longitudinal pore water Darcy velocity, m/s. u* normalized longitudinal velocity, equal to u/u,,,. u song underflow velocity (longitudinal porewater Darcy velocity due to hydraulic gradient), equal to KS, m/s. u * normalized underflow velocity, equal to U ong/u,,. long u,, pore water Darcy velocity scale, equal to Kkhm,' m/s. v vertical Darcy velocity, m/s. v* normalized vertical velocity, equal to V/Urn. X horizontal coordinate, m. x* normalized horizontal coordinate, equal to kx. X horizontal coordinate of a streamline, m. y vertical coordinate, m. y* normalized vertical coordinate, equal to ky. Y vertical coordinate of a streamline, m. a ratio of transverse to longitudinal dispersion. rl bed surface elevation, m. 0 porosity of the bed material. X bed form wavelength, m. rr r.m.s. bed elevation, m. r time lag, s. Acknowledgments. This study was supported under USGS grant G1488, the Andrew W. Mellon Foundation, and a Walter L. and Reta Mae Moore Fellowship. The authors thank James J. Morgan and Robert C. Y. Koh for their advice on this project. The authors also thank the Water Resources Research reviewers of the original manuscript for their detailed and helpful comments. References Bear, A., and A. Verruijt, Modelling of Groundwater Flow and Pollution, Kluwer Acad., Norwell, Mass., Bear, J., Dv. namics of Fhdds in Porous Media, Elsevier, New York, Bencala, K. 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